Add proximity loops to sharp edge without affecting mesh shape
You could select those edges and bevel them using the interactive bevel tool (CtrlB) and move your mouse to adjust the level. This will create an additional set of faces that you also ring cut to get an even sharper edge without affecting the shape of the object.

Note that adding additional ring cuts to this face will create a pinch in the corners due to the clump of vertices on the side so you might have to use it sparingly depending on any other edges you selected.